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In this paper, we consider the upper bound of the probabilistic stardiscrepancy based on Hilbert space filling curve sampling. This problemoriginates from the multivariate integral approximation, but the main resultremoves the strict conditions on the sampling number of the classicalgrid-based jittered sampling. The main content has three parts. First, weinherit the advantages of this new sampling and achieve a better upper bound ofthe random star discrepancy than the use of Monte Carlo sampling. In addition,the convergence order of the upper bound is improved from $O(N^{-\frac{1}{2}})$to $O(N^{-\frac{1}{2}-\frac{1}{2d}})$. Second, a better uniform integralapproximation error bound of the function in the weighted space is obtained.Third, other applications will be given. Such as the sampling theorem inHilbert spaces and the improvement of the classical Koksma-Hlawka inequality.Finally, the idea can also be applied to the proof of the strong partitionprinciple of the star discrepancy version.

Star discrepancy bounds based on Hilbert space filling curve stratified sampling and its applications